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Venn Diagrams (how they work) and Sorting Polygons

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4.14 properties of 2D shapes - Basic Maths Core Skills Level 4

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Quadrilateral just means "four sides"
( quad means four, lateral means side).

A Quadrilateral has four-sides, it is 2-dimensional (a flat shape), closed (the lines join up), and has straight sides.

Try it Yourself

geometry/images/geom-quad.js?mode=choose
(Also see this on Interactive Quadrilaterals)
Properties

A quadrilateral has:

four sides (edges)

four vertices (corners)

interior angles that add to 360 degrees:

Try drawing a quadrilateral, and measure the angles. They should add to 360°

Types of Quadrilaterals

There are special types of quadrilateral:

Some types are also included in the definition of other types! For example a square, rhombus and rectangle are also parallelograms. See below for more details.

Let us look at each type in turn:

The Rectangle

the little squares in each corner mean "right angle"
A rectangle is a four-sided shape where every angle is a right angle (90°).

Also opposite sides are parallel and of equal length.

the little squares in each corner mean "right angle"
The Square

A square has equal sides (marked "s") and every angle is a right angle (90°)

Also opposite sides are parallel.

A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length).

The Rhombus

A rhombus is a four-sided shape where all sides have equal length (marked "s").

Also opposite sides are parallel and opposite angles are equal.

Another interesting thing is that the diagonals (dashed lines) meet in the middle at a right angle. In other words they "bisect" (cut in half) each other at right angles.

A rhombus is sometimes called a rhomb or a diamond.

The Parallelogram

A parallelogram has opposite sides parallel and equal in length. Also opposite angles are equal (angles "A" are the same, and angles "B" are the same).

NOTE: Squares, Rectangles and Rhombuses are all Parallelograms!

Example:

A parallelogram with:

all sides equal and

angles "A" and "B" as right angles

is a square!

The Trapezoid (UK: Trapezium)

A trapezoid (called a trapezium in the UK) has a pair of opposite sides parallel.

And a trapezium (called a trapezoid in the UK) is a quadrilateral with NO parallel sides:

An Isosceles trapezoid, as shown above, has left and right sides of equal length that join to the base at equal angles.

The Kite

Hey, it looks like a kite (usually).

It has two pairs of sides:

Each pair is made of two equal-length sides that join up.

Also:

the angles where the two pairs meet are equal.

the diagonals, shown as dashed lines above, meet at a right angle.

one of the diagonals bisects (cuts equally in half) the other.

... and that's it for the special quadrilaterals.

Irregular Quadrilaterals

The only regular (all sides equal and all angles equal) quadrilateral is a square. So all other quadrilaterals are irregular.

The "Family Tree" Chart

Quadrilateral definitions are inclusive.
(We don't say "Having all 90° angles makes it a rectangle except when all sides are equal then it is a square.")
Example: a square is also a rectangle.

So we include a square in the definition of a rectangle.

This may seem odd, as in daily life we think of a square as not being a rectangle ... but in mathematics it is.

Using the chart below we can answer such questions as:

Is a Square a type of Rectangle? (Yes)

Is a Rectangle a type of Kite? (No)

Complex Quadrilaterals

Oh Yes! when two sides cross over, we call it a "Complex" or "Self-Intersecting" quadrilateral, like these:

They still have 4 sides, but two sides cross over.

Polygon

A quadrilateral is a polygon. In fact it is a 4-sided polygon, just like a triangle is a 3-sided polygon, a pentagon is a 5-sided polygon, and so on.

Play with Them

Now that you know the different types, you can play with the Interactive Quadrilaterals.

Other Names

A quadrilateral can sometimes be called:

a Quadrangle ("four angles"), so it sounds like "triangle"

a Tetragon ("four polygon"), so it sounds like "pentagon", "hexagon", etc.

621,622,623,624,763,764, 2128, 2129, 3230, 3231

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Quadrilateral just means four sides(quad means four, lateral means side). ... A Quadrilateral has four-sides, it is 2-dimensional (a flat shape), closed (the lines join up), and

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Intro to Symmetry: All About Symmetry for Kids - FreeSchool

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Polygon

A polygon is a plane shape (two-dimensional) with straight sides. Examples include triangles, quadrilaterals, pentagons, hexagons and so on.

Regular

A "Regular Polygon" has:

Otherwise it is irregular.

Regular Pentagon

Irregular Pentagon

Here we look at Regular Polygons only.

Properties

So what can we know about regular polygons? First of all, we can work out angles.

All the Exterior Angles of a polygon add up to 360°, so:

Each exterior angle must be 360°/n

(where n is the number of sides)

Press play button to see.

Exterior Angle
(of a regular octagon)

Example: What is the exterior angle of a regular octagon?

An octagon has 8 sides, so:

Interior Angle = 180° − Exterior Angle

We know the Exterior angle = 360°/n, so:

Interior Angle = 180° − 360°/n

Which can be rearranged like this:

So we also have this:

Interior Angle = (n−2) × 180° / n

Example: What is the interior angle of a regular octagon?

A regular octagon has 8 sides, so:

Exterior Angle = 360 ° / 8 = 45°

Interior Angle = 180° − 45° = 135°

Interior Angle
(of a regular octagon)
Or we could use:

Example: What are the interior and exterior angles of a regular hexagon?

A regular hexagon has 6 sides, so:

Exterior Angle = 360 ° / 6 = 60°

Interior Angle = 180 ° − 60° = 120°

And now for some names:

"Circumcircle, Incircle, Radius and Apothem ..."

Sounds quite musical if you repeat it a few times, but they are just the names of the "outer" and "inner" circles (and each radius) that can be drawn on a polygon like this:

The "outside" circle is called a circumcircle, and it connects all vertices (corner points) of the polygon.

The radius of the circumcircle is also the radius of the polygon.

The "inside" circle is called an incircle and it just touches each side of the polygon at its midpoint.

The radius of the incircle is the apothem of the polygon.

(Not all polygons have those properties, but triangles and regular polygons do).

Breaking into Triangles

We can learn a lot about regular polygons by breaking them into triangles like this:

Notice that:

the "base" of the triangle is one side of the polygon.

the "height" of the triangle is the "Apothem" of the polygon

Now, the area of a triangle is half of the base times height, so:

Area of one triangle = base × height / 2 = side × apothem / 2

To get the area of the whole polygon, just add up the areas of all the little triangles ("n" of them):

Area of Polygon = n × side × apothem / 2

And since the perimeter is all the sides = n × side, we get:

Area of Polygon = perimeter × apothem / 2

A Smaller Triangle

(Note: The angles are inradians, not degrees)
By cutting the triangle in half we get this:

The small triangle is right-angled and so we can use sine, cosine and tangent to find how the side, radius, apothem and n (number of sides) are related:

sin(π/n) = (Side/2) / Radius
Side = 2 × Radius × sin(π/n)

cos(π/n) = Apothem / Radius
Apothem = Radius × cos(π/n)

tan(π/n) = (Side/2) / Apothem
Side = 2 × Apothem × tan(π/n)

There are a lot more relationships like those (most of them just "re-arrangements"), but those will do for now.

More Area Formulas

We can use that to calculate the area when we only know the Apothem:

And we know (from the "tan" formula above) that:

Side = 2 × Apothem × tan(π/n)

So:

And there are 2 such triangles per side, or 2n for the whole polygon:

Area of Polygon = n × Apothem ² × tan(π/n)

When we don't know the Apothem, we can use the same formula but re-worked for Radius or for Side:

Area of Polygon = ½ × n × Radius ² × sin(2 × π/n)

Area of Polygon = ¼ × n × Side ² / tan(π/n)

A Table of Values

And here is a table of Side, Apothem and Area compared to a Radius of "1", using the formulas we have worked out:

Type

Name when
Regular

Sides
(n)

Shape

Interior Angle

Radius

Side

Apothem

Area

Triangle
(or Trigon)

Equilateral
Triangle
3
60°

1

1.732
(√3)

0.5

1.299
(¾√3)

Quadrilateral
(or Tetragon)4
Square

90°

1

1.414
(√2)

0.707
(1/√2)

2

Pentagon
Regular
Pentagon
5
108°

1

1.176

0.809

2.378

Hexagon
Regular
Hexagon
6
120°

1

1

0.866
(½√3)

2.598
((3/2)√3)

Heptagon
(or Septagon)

Regular
Heptagon
7
128.571°

1

0.868

0.901

2.736

Octagon
Regular
Octagon
8
135°

1

0.765

0.924

2.828
(2√2)

...

...

Pentacontagon

Regular
Pentacontagon
50
172.8°

1

0.126

0.998

3.133

(Note: values correct to 3 decimal places only)

Graph

And here is a graph of the table above, but with number of sides ("n") from 3 to 30.

Notice that as "n" gets bigger, the Apothem is tending towards 1 (equal to the Radius) and that the Area is tending towards π = 3.14159..., just like a circle.

What is the Side length tending towards?

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A polygon is a plane shape (two-dimensional) with straight sides. Examples include triangles, quadrilaterals, pentagons, hexagons and so on. A "Regular Polygon" has: Otherwise it is irregular. Regular Pentagon Irregular Pentagon Here we look at Regular Polygons only. So what can we know about regular polygons? First of all, we can work out angles.

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Lines of Symmetry: Grade 4 Module 4 Lesson 12

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DIFFERENCE BETWEEN SIMILAR & CONGRUENT FIGURES?

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Regular Polygons

A polygon is a plane (2D) shape with straight sides.
To be a regular polygon all the sides and angles must be the same:

Heptagon - 7 Sides

Nonagon - 9 Sides

Decagon - 10 Sides

More ...

Other Common Polygons

These are 2D shapes, and they are polygons, but not regular polygons:

Curved Shapes

These are also 2D shapes, but are not polygons because they have curves:

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Regular Polygons A polygon is a plane (2D) shape with straight sides. To be a regular polygon all the sides and angles must be the same: Heptagon - 7 Sides Nonagon - 9 Sides Decagon - 10 Sides More ...

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The Polygon Song | Polygons for Kids | Polygons Geometry | Silly School Songs

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